Branch-Curvature Mass Closure for the Higgs Sector and a β-Dominated Scalar Mode

This technical manuscript formulates a common effective mass-closure logic for two distinct scalar sectors within the alpha-beta-phi framework: the Higgs-sector closure near 125 GeV and a beta-dominated structural scalar mode near 285 GeV. The core thesis is that scalar masses in this construction are defined by the local curvature of effective branch potentials after canonical normalization and diagonalization, rather than being assigned through an infinite chain of mass-giving particles. Key Contributions:1. Higgs Sector Closure: The manuscript retains the standard spontaneous-symmetry-breaking potential but promotes its coefficients to branch-dependent effective quantities. Within a specific "q=2" anchor, it establishes a controlled algebraic relation between the Higgs mass and the top quark mass, yielding a benchmark value of approximately 125.10 GeV. 2. Structural Scalar Mode: The research identifies a second scalar excitation originating from the structural channel (beta). By separating the dimensionless hierarchy variable from a canonically normalized field, the physical mass is determined through the diagonalization of the mixed scalar sector, placing a stable benchmark near 285 GeV. 3. Mass-Origin Principle: The framework provides a field-theoretic answer to the origin of scalar masses by identifying them as the second derivative (Hessian) of the effective potential at the relevant electroweak or structural branch. 4. Scope and Limitations: The results are presented as an effective mass-closure architecture rather than a final ultraviolet derivation. The manuscript distinguishes these scalar modes from dark matter and dark energy, treating the latter as effective projections of a filtered reservoir rather than identifying them directly with the 285 GeV mode. This work establishes a mathematically explicit domain for scalar masses within a structured vacuum, providing a foundation for future microscopic modeling and collider phenomenology.